Diophantine approximation with primes from short intervals

In this paper, we establish hybrid results on Diophantine approximation with primes from short intervals. In particular, we prove the following result in a slightly modified form: If α is an irrational number having a continued fraction expansion with bounded terms (in particular, if α is a quadratic irrational), then the number of primes p in the interval (X – Y,X] satisfying ∥pα∥ < δ is asymptotically equal to 2δY/log X, provided that X ≥ 10, X 2/3+ε ≤ Y ≤ X/2 and X ε max {X 1/4 Y −1/2 ,X 2/3 Y −1 } ≤ δ ≤ 1/2.